Simplify and expand the following expression: $ \dfrac{n}{n - 7}+\dfrac{n - 4}{n - 10} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(n - 7)(n - 10)$ Multiply the first term by $\dfrac{n - 10}{n - 10}$ $ \begin{align*} \dfrac{n}{n - 7} \times \dfrac{n - 10}{n - 10} & = \dfrac{(n)(n - 10)}{(n - 7)(n - 10)} \\ & = \dfrac{n^2 - 10n}{(n - 7)(n - 10)}\end{align*} $ Multiply the second term by $\dfrac{n - 7}{n - 7}$ $ \begin{align*} \dfrac{n - 4}{n - 10} \times \dfrac{n - 7}{n - 7} & = \dfrac{(n - 4)(n - 7)}{(n - 10)(n - 7)} \\ & = \dfrac{n^2 - 11n + 28}{(n - 10)(n - 7)}\end{align*} $ Now we have: $ = \dfrac{n^2 - 10n}{(n - 7)(n - 10)} + \dfrac{n^2 - 11n + 28}{(n - 10)(n - 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{n^2 - 10n + n^2 - 11n + 28}{(n - 7)(n - 10)} $ $ = \dfrac{2n^2 - 21n + 28}{(n - 7)(n - 10)}$ Expand the denominator: $ = \dfrac{2n^2 - 21n + 28}{n^2 - 17n + 70}$